# American Institute of Mathematical Sciences

October  2011, 5(4): 747-769. doi: 10.3934/jmd.2011.5.747

## Partially hyperbolic diffeomorphisms with compact center foliations

 1 Department ofMathematical Sciences, The State University of New York, Binghamton, NY 13902, United States

Received  November 2011 Revised  February 2012 Published  March 2012

Let $f\colon M\to M$ be a partially hyperbolic diffeomorphism such that all of its center leaves are compact. We prove that Sullivan's example of a circle foliation that has arbitrary long leaves cannot be the center foliation of $f$. This is proved by thorough study of the accessible boundaries of the center-stable and the center-unstable leaves.
Also we show that a finite cover of $f$ fibers over an Anosov toral automorphism if one of the following conditions is met:
1. 1. the center foliation of $f$ has codimension 2, or
2. 2. the center leaves of $f$ are simply connected leaves and the unstable foliation of $f$ is one-dimensional.
Citation: Andrey Gogolev. Partially hyperbolic diffeomorphisms with compact center foliations. Journal of Modern Dynamics, 2011, 5 (4) : 747-769. doi: 10.3934/jmd.2011.5.747
##### References:
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##### References:
 [1] V. M. Alekseev and M. V. Yakobson, Symbolic dynamics and hyperbolic dynamic systems, Phys. Rep., 75 (1981), 287-325. doi: 10.1016/0370-1573(81)90186-1.  Google Scholar [2] D. Bohnet, "Partially Hyperbolic Systems with a Compact Center Foliation with Finite Holonomy," Thesis, 2011. Google Scholar [3] C. Bonatti and A. Wilkinson, Transitive partially hyperbolic diffeomorphisms on 3-manifolds, Topology, 44 (2005), 475-508. doi: 10.1016/j.top.2004.10.009.  Google Scholar [4] R. Bowen, Periodic points and measures for axiom A diffeomorphisms, Trans. Amer. Math. Soc., 154 (1971), 377-397.  Google Scholar [5] M. Brin, D. Burago and S. Ivanov, On partially hyperbolic diffeomorphisms of 3-manifolds with commutative fundamental group, in "Modern Dynamical Systems and Applications," 307-312, Cambridge Univ. Press, Cambridge, 2004.  Google Scholar [6] D. Burago and S. Ivanov, Partially hyperbolic diffeomorphisms of 3-manifolds with abelian fundamental groups, J. Mod. Dyn., 2 (2008), 541-580. doi: 10.3934/jmd.2008.2.541.  Google Scholar [7] A. Candel and L. Conlon, "Foliations. I," Graduate Studies in Mathematics, 23, American Mathematical Society, Providence, RI, 2000.  Google Scholar [8] P. Carrasco, "Compact Dynamical Foliations," Thesis, 2010. Google Scholar [9] Y. Coudene, Pictures of hyperbolic dynamical systems, Notices Amer. Math. Soc., 53 (2006), 8-13.  Google Scholar [10] R. Edwards, K. Millett and D. Sullivan, Foliations with all leaves compact, Topology, 16 (1977), 13-32. doi: 10.1016/0040-9383(77)90028-3.  Google Scholar [11] D. B. A. Epstein, Periodic flows on three-manifolds, Ann. of Math. (2), 95 (1972), 66-82. doi: 10.2307/1970854.  Google Scholar [12] D. B. A. Epstein, Foliations with all leaves compact, Ann. Inst. Fourier (Grenoble), 26 (1976), 265-282. doi: 10.5802/aif.607.  Google Scholar [13] D. B. A. Epstein and E. Vogt, A counterexample to the periodic orbit conjecture in codimension $3$, Ann. of Math. (2), 108 (1978), 539-552. doi: 10.2307/1971187.  Google Scholar [14] J. Franks, Anosov diffeomorphisms, in "1970 Global Analysis" (Proc. Sympos. Pure Math., Vol. XIV, Berkeley, Calif., 1968), Amer. Math. Soc., Providence, R.I., 61-93.  Google Scholar [15] K. Hiraide, A simple proof of the Franks-Newhouse theorem on codimension-one Anosov diffeomorphisms, Ergodic Theory Dynam. Systems, 21 (2001), 801-806. doi: 10.1017/S0143385701001390.  Google Scholar [16] J. G. Hocking and G. S. Young, "Topology," Second edition, Dover Publications, Inc., New York, 1988.  Google Scholar [17] R. Langevin, A list of questions about foliations, in "Differential Topology, Foliations, and Group Actions" (Rio de Janeiro, 1992), Contemp. Math., 161, Amer. Math. Soc., Providence, RI, (1994), 59-80.  Google Scholar [18] D. Montgomery, Pointwise periodic homeomorphisms, Amer. J. Math., 59 (1937), 118-120. doi: 10.2307/2371565.  Google Scholar [19] S. E. Newhouse, On codimension one Anosov diffeomorphisms, Amer. J. Math., 92 (1970), 761-770. doi: 10.2307/2373372.  Google Scholar [20] F. Rodriguez Hertz, M. A. Rodriguez Hertz and R. Ures, A survey of partially hyperbolic dynamics, in "Partially Hyperbolic Dynamics, Laminations, and Teichmüler Flow," Fields Inst. Commun., 51, Amer. Math. Soc., Providence, RI, (2007), 35-87. Google Scholar [21] , F. Rodriguez Hertz, M. A. Rodriguez Hertz and R. Ures,, private communication., ().   Google Scholar [22] D. Sullivan, A counterexample to the periodic orbit conjecture, Inst. Hautes Études Sci. Publ. Math., 46 (1976), 5-14.  Google Scholar [23] E. Vogt, Foliations of codimension 2 with all leaves compact, Manuscripta Math., 18 (1976), 187-212. doi: 10.1007/BF01184305.  Google Scholar
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